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We generalize our earlier results concerning meshfree collocation methods forsemilinear elliptic second order problems to the quasilinear case. Thestability question, however, is treated differently, namely by extending apaper on uniformly stable discretizations of well-posd linear problems to thenonlinear case. These two ingredients allow a proof that all well-posedquasilinear elliptic second-order problems can be discretized in a uniformlystable way by using sufficient oversampling, and then the error of thenumerical solution behaves like the error obtainable by direct approximation ofthe true solution by functions from the chosen trial space, up to a factorinduced by being forced to use a H\"older-type theory for the nonlinear PDE. Weapply our general technique to prove convergence of meshfree methods forquasilinear elliptic equations with Dirichlet and non-Dirichlet boundaryconditions. This is achieved for bifurcation and center manifolds of ellipticpartial differential equations and their numerical methods as well.

A Nonlinear Discretization Theory for Meshfree Collocation Methods applied to Quasilinear Elliptic Equations